I've just finished teaching a year-long "foundations of algebraic
geometry" class. It
was my third time teaching it, and my notes are gradually converging.
I've enjoyed it for a number of reasons (most of all the students, who
were smart, hard-working, and from a variety of fields). I've
particularly enjoyed talking with experts (some in nearby fields, many
active on mathoverflow) about what one should (or must!) do in a first
schemes course. I've been pleasantly surprised to find that those who
have actually thought about teaching such a course (and hence who know
how little can be covered) tend to agree on what is important, even if
they are in very different parts of the subject. I want to raise this
question here as well:

What topics/examples/ideas etc. really really should be learned in a
year-long first serious course in schemes?

Here are some constraints. Certainly most excellent first courses
ignore some or all of these constraints, but I include them to focus
the answers. The first course in question should be
purely algebraic. (The reason for this constraint: to avoid a
debate on which is the royal road to algebraic geometry --- this is
intended to be just one way in. But if the community thinks that a
first course should be broader, this will be reflected in the voting.)
The course should be intended for people in all parts of algebraic
geometry. It should attract smart people in nearby areas. It should
not get people as quickly as possible into your particular area of
research. Preferences: It can (and, I believe, must) be hard. As
much as possible, essential things must be proved, with no handwaving
(e.g. "with a little more work, one can show that...", or using
exercises which are unreasonably hard). Intuition should be given
when possible.

Why I'm asking: I will likely edit the notes further, and hope to post
them in chunks over the 2010-11 academic year to provoke further debate. Some hastily-written thoughts are
here,
if you are curious.

As usual for big-list questions: one topic per answer please. There
is little point giving obvious answers (e.g. "definition of a
scheme"), so I'm particularly interested in things you think others
might forget or disagree with, or things often omitted, or things you
wish someone had told you when you were younger. Or propose dropping
traditional topics, or a nontraditional ordering of traditional topics. Responses
addressing prerequisites such as "it shouldn't cover any commutative
algebra, as participants should take a serious course in that subject
as a prerequisite" are welcome too. As the most interesting
responses might challenge (or defend) conventional wisdom, please give
some argument or evidence in favor of your opinion.

Update later in 2010: I am posting the notes, after suitable editing, and trying to take into account the advice below, here. I hope to reach (near) the end some time in summer 2011. Update July 2011: I have indeed reached near the end some time in summer 2011.

Dear Ravi, while I'm not sure if this should be taught in a first schemes course, but it's something that I'd love to see exposited more fully. Jim Borger gave an outline of a program to jump straight into algebraic spaces, skipping schemes entirely. Maybe you could figure out a way to do it? sbseminar.wordpress.com/2009/08/06/…
–
Harry GindiJun 17 '10 at 12:57

A parsing question: does "first serious schemes course" mean that there could be a prior, not-so-serious course on schemes? Or do you mean "first, serious schemes course"?
–
Pete L. ClarkJun 17 '10 at 17:15

3

Re: community wiki. Ravi considered this (see the meta thread mentioned above), and decided against it, so I'm not going to use the wiki-hammer unless asked.
–
Scott Morrison♦Jun 17 '10 at 19:42

31 Answers
31

One of the wholly unnecessary reasons that schemes are regarded with such
fear by so many mathematicians in other fields is that three, largely
orthogonal, generalizations are made simultaneously.

Considering a "variety" to be Spec or Proj of a domain finitely
generated over an algebraically closed field, the generalizations are
basically

Allowing nilpotents in the ring

Gluing affine schemes together

Working over a base ring that isn't an algebraically closed field
(or even a field at all).

For many years I got by with only #1. More recently I've been interested
in #1 + #3. Presumably someday I'll care about #2, but not yet.
Anyway I think it's crazy to give the impression that the three are a
package deal that one must buy all of simultaneously, rather than in
much easier installments.

I think it could be useful to explain which subfield of mathematics, or which important example, motivates which of #1,#2,#3 is really a necessary generalization.

The way I think about it is: #1 is analysis (looking at near-solutions $P_1(x)=\ldots=P_k(x)=O(\varepsilon)$ of equations instead of exact equations $P_1(x)=\ldots=P_k(x)=0$). #2 is differential geometry (looking at manifolds instead of coordinate patches). #3 is number theory (solving equations over number fields, rings of integers, etc.). This way I can minimise my exposure to algebra and topology, my two weakest suits. :-)
–
Terry TaoJun 18 '10 at 5:57

5

Allen, in what sense do you mean you've never cared about #2? I'd have thought it is the basic ingredient by which one thinks about any non-affine object (akin to manifolds, as in Terry's comment), and so much more intuitive than #1 and #3? I assume you are implicitly speaking of gluing in the Zariski topology rather than the etale topology (algebraic spaces...), but perhaps I misunderstand.
–
BCnrdJun 18 '10 at 7:01

4

I like these three a lot. In each case things the new notion is forced upon you by nature. I like Terry's interpretation (and like to note that certain notions are "geometry" or "arithmetic" --- I'd never used the word "analysis" for similar reasons to Terry's reasons for algebra and topology... :-). What about #4: non-closed points? This could even bump out #2, which is already present in Proj. Somehow this local-to-global issue is already present in how one thinks of a manifold (and even if people may not know it well at this stage, they have a good intuitive idea of it).
–
Ravi VakilJun 18 '10 at 13:22

As you should, you prove the Nullstellensatz early on, as the statement that the closed points of $\mathbb{A}^n_k$ are in bijection with $k^n$, for $k$ an algebraically closed field. I wonder whether it is also a good idea to say that, for any $k$, the closed points of $\mathbb{A}^n_k$ are in bijection with the Galois orbits in $\overline{k}^n$. This might require too big a digression into Galois theory, but I remember a number of my grad school classmates having confusions about closed points over non-algebraically closed fields which could be immediately answered from this decsription.

I definitely second this. I understood $\mathrm{Spec}\mathbb{R}[x]$ when someone said that it's just $\mathbb{C}$ modulo conjugation. And I think it's fair to assume that people learning schemes have some Galois theory from grad algebra (just like you assume they know what a topological space is, because if they don't, things have gone horribly awry)
–
Charles SiegelJun 17 '10 at 15:30

1

David, great ideas.I currently state the Nullst. early on, but because the proof will fall out of things later (Noether normalization), I hold off proving it. I definitely do the Galois-conjugate thing --- if you don't know what the points are of a space, you can get very confused! (Aside: it is interesting that we can quickly describe the primes in $\mathbb{C}[x_1,\dots,x_n]$ for $n=0,1,2$, but for $n=3$ there are weirder sorts of primes --- and there is no way of not thinking of them in terms of geometry.) Charles, doing the explicit case of $\mathbb{R}[x]$ makes lots of sense.
–
Ravi VakilJun 17 '10 at 23:50

7

@Harry: the suggestion to view Zariski's Lemma as a corollary of Zariski's Main Theorem sounds dangerously close to (if not actually) being circular, since I can't imagine developing algebraic geometry even remotely far enough to get to ZMT without already knowing the Nullstellensatz (which is more or less equivalent to Zariski's Lemma, depending on how one defines "Nullstellensatz").
–
BCnrdJun 18 '10 at 7:28

I found differentials hard to understand when I learned this material. Here are two things that helped me which I think are not in your notes:

(1) The description of the Zariski tangent space to $X$ at $x$ as those Hom's from $\mathrm{Spec} \ k[\epsilon]/\epsilon^2$ to $X$ which take $\mathrm{Spec} \ k$ to $x$. This is this is much closer to my physical intuition for a tangent space than the $(\mathfrak{m}/\mathfrak{m}^2)^{\vee}$ definition. It is also an early example of the power of using rings with nilpotents. Building the vector space structure from this definition is especially pretty.

(2) A careful discussion of the relationship between the infinitesimal objects, i.e. the elements of the Zariski tangent and cotangent spaces, and the global objects, i.e. derivations and Kahler differentials.

Harry: that's farther than I can manage to get in a year without losing people, although it is possible to foreshadow the dualizing complex (as a side remark) and even essentially to define it.
–
Ravi VakilJun 17 '10 at 23:45

Let me begin with something essentially obvious: students should learn to work with non-closed points. In practice, this means learning how to use them to simplify life.

Here are some suggestions as to how to do that:

(a) Explain that coherent sheaves are generically free, and use this to prove things
like generic smoothness of varieties (by applying it to the tangent sheaf).

(b) Explain carefully the proof of Chevalley's theorem that the image of constructible
is contstructible. (Note that this latter result has the advantage of being extremely
useful, and also has likely not been covered in any form in a previous varieties course.)

Note also that one can deduce the Nullstellensatz from this result, which kills two
birds with one stone. (See the discussion in this answer, and the notes of Mumford and Oda that are linked there.)

(c) one can beef up (a) by looking at say a fibration $X \to Y,$ and then looking at fibres
over a generic point of $Y$, and then extending information to a n.h. of that point.
Incidentally, it was the desirability of this kind of argument that first led Zariski to point out the importance of studying algebraic geometry over non-algebraically closed fields. For him, these non-algebraically closed fields were not $\mathbb Q$ or $\mathbb F_p$, but rather function fields of varieties (with the initial ground field being a good old fashioned algebraically closed field).

Examples like this last one can really help demystify not just the role of generic points, but also the role of non-algebraically closed fields. (In particular, they show that the latter are not just of interest in number theory. Zariski was certainly not a number theorist!)

Just to add to Emerton's remarks, it good to note that the scheme-theoretic results on open/closed/constructible sets are stronger than variety counterparts. If one knows a constructibility result for a subset of an integral scheme of finite type over an alg. closed field (e.g., locus defined by conditions on fibers of morphism) and can prove generic point lies in there then a classical Zariski-dense open locus of classical points are in there too. This is what underlies useful instances of the "spreading out" principle in (c). This comes up for rigid-analytic vs. Berkovich spaces too!
–
BCnrdJun 18 '10 at 14:49

3

I agree very much; after taking the class, I still wish I understood generic points better. They make proofs so much easier, in a way unique to algebraic geometry, but I still don't feel that I have a good enough handle to use them. One of my main points of confusion: how should I think of the generic fiber? stalk at a generic point? What's the difference? Also, I still feel unsure of how good the analogy between properties at the generic point and generic properties from differential topology is.
–
Ilya GrigorievJun 18 '10 at 19:23

3

Ilya, drop by my office (or Ravi's) some time and we can clear it up. The name "generic point" is very much deserved. As to why, this is one of those things which is extremely well-developed in EGA and almost invisible in Hartshorne.
–
BCnrdJun 18 '10 at 20:42

3

Ravi, what helps is to show the "spreading out" principle in practice with multiple examples that can be seen by hand, and mention of others which are clearly much deeper (in the sense of not being obviously captured by equations) but still reassuring to know about. And also counterexamples (such as irreducible vs. absolutely irreducible fibers, as you know). The example of vanishing of a function is a bit too simple to convey the flavor of how such things go. A bare-hands proof of generic freeness for coherent sheaves via linear algebra at the generic point is a better first example.
–
BCnrdJun 19 '10 at 0:44

1

I'm happy with that example; it may as well be a second example, and still mention the first.
–
Ravi VakilJun 20 '10 at 20:48

Since in 2007-2008 you evoked [ Class 24, §1.8, The problem with locally free sheaves] the equivalence between locally free sheaves and vector bundles on a scheme, the following point, potentially confusing for a beginner, could be mentioned.

A locally free sheaf $\mathcal E$ has a sheaf fibre $\mathcal E_x$ at $x$ but also a vector fibre $\mathcal E[x]=\mathcal E_x \otimes _ {\mathcal O_x} k(x)$. The fact that tensoring is not exact explains the paradox that a locally free subsheaf of a locally free sheaf does not yield a sub-vector bundle of a vector bundle in the above equivalence. The contrasting notation $\mathcal E[x]$ versus $\mathcal E_x$ (that I learned from German mathematicians) may help clarify this subtle point .

I am quite aware that there is nothing grandiose in this technical suggestion, but little points like those can be quite frustrating when learning a new subject

This is actually fantastic. (I think the devil in algebraic geometry really is in the details, and things like this make a big difference in assuaging confusion.) It is indeed confusing that the subscript $x$ is used for both stalks and for fibers. Am I understanding you correctly that you would propose that if $\mathcal{F}$ is a quasicoherent sheaf on $X$, that $\mathcal{F}[x]$ be used for the fiber over a point $x \in X$? This would prevent a lot of confusion.
–
Ravi VakilJun 18 '10 at 0:08

4

I always write $\mathcal{E}(x)$ for the "locally ringed space fiber", and (naively) thought it was universal notation nowadays (not sure where I got it from). I agree that a clear distinction of fiber and stalk can be a puzzler early in the education process. Sadly, Godement's book uses the exact same notation for stalks! (Neither Godement nor EGA seem to have a special notation for locally ringed space fiber: each explicitly writes out the monstrosity $\mathcal{E}_x/\mathfrak{m}_x \mathcal{E}_x$ every time it is needed.)
–
BCnrdJun 18 '10 at 0:43

2

$\mathcal F|_x$ is another possibility which avoids confusion with twists and shifts.
–
user2035Jun 18 '10 at 6:58

13

Ravi, the reason I advocate $\mathcal{F}(x)$ for the fiber at $x$ is that just as one writes $s_x$ for the $x$-stalk (in $\mathcal{F}_x$) of a local section $s$ of $\mathcal{F}$, it is nice to write $s(x)$ for the image of $s_x$ in $\mathcal{F}(x) := \mathcal{F}_x/\mathfrak{m}_x \mathcal{F}_x$ because that is suggestive of "evaluating a function" (which literally is what happens when $\mathcal{F} = O_X$!). In other words, the $\mathcal{F}(x)$ notation reminds one of evaluation, which is the difference between "stalk" and "fiber". We can battle it out further in your office later today...
–
BCnrdJun 18 '10 at 14:25

3

Ravi, please do not write $\mathcal{O}(x)$ for residue field; I hope it was meant as a joke. Stick with $k(x)$ or $\kappa(x)$ for the residue field. I don't see why getting rid of $\kappa(x)$ is viewed as an "advantage". It is the standard notation in EGA, for example, and sure looks more like a field than $\mathcal{O}(x)$ (as if one knows what a field "looks like").
–
BCnrdJun 18 '10 at 20:38

Toric varieties. They're so easy to define and work with, and to organize examples around. Like blowing up a scheme at a fat point, or blowing up in different orders, or big but not ample line bundles, ... Of course there's the danger that they'll give people the wrong idea about what general schemes are like, but a few curves-of-high-genus examples should help with that.

This is only indirectly related (because "varieties not schemes"). How can a complete variety fail to be projective? I only understood this once I learned toric varieties.
–
Victor ProtsakJun 18 '10 at 0:13

1

Victor, I think that's quite related, and a good point. I currently do an easy surface example (again requiring flatness), but toric varieties would provide a very pleasant other example.
–
Ravi VakilJun 18 '10 at 0:18

3

I like taking an octahedron, splitting it into northern and southern hemispheres glued along a neighborhood of an equatorial square, and stretching out the top vertex into an interval. Then when you try to glue these two together, the geometry of the top half wants the equator to be a rectangle, but the bottom half requires it to be a square. Hence any line bundle will be degree 0 on the P^1 corresponding to this new top edge.
–
Allen KnutsonJun 18 '10 at 13:41

There is a very useful and simple lemma on sheaves which is (I think) never explicitly stated in Hartshorne. It is Proposition I-12 of Eisenbud-Harris. I think you should definitely make sure to explicitly state this. Sheafification was very scary and mysterious to me until I learned this lemma.

The fact that this is not stated in Hartshorne is one of the reasons why his construction of the structure sheaf of an affine scheme is so ad-hoc.
–
Harry GindiJun 17 '10 at 17:59

6

My first (limited) exposure to schemes was in a course from Joe Harris which used a draft copy of the book with Eisenbud, and most of which went completely over my head. But when I eventually "graduated" to reading Hartshorne I knew to ignore his mysterious construction of the structure sheaf and followed the "B-sheaf" method from Eisenbud-Harris instead, with the help of Exercise 23 in Chapter 3 of Atiyah-MacDonald.
–
BCnrdJun 17 '10 at 18:25

4

Wow, a reference for that lemma! Thanks, Kevin.
–
David SpeyerJun 17 '10 at 18:48

6

I think Chapter I of Eisenbud-Harris is really great in general.
–
Kevin H. LinJun 17 '10 at 21:12

5

On a related note (also related to Allen's discussion of schemes as gluing together affines): I have found that using the fact that schemes are affines glued together, rather than just ringed spaces, makes showing facts we care about regarding quasicoherent sheaves much easier. In particular, in graduate school, it was a revelation when a visiting grad school (who now goes under the pseudonym "BCnrd") pointed out to me the simple fact that the intersection of 2 affines is a union of affines simultaneously distinguished in them both. This turned a host of Hartshorne ideas from hard to easy.
–
Ravi VakilJun 18 '10 at 13:38

Dear Ravi, here is a small suggestion. I think one might emphasize as soon as possible that the subschemes of an affine scheme $Spec A$ exactly correspond to the set of ideals of the ring $A$.( I don't know if this is deep or tautological: probably both.) This allows one to illustrate many of the strange and frightening features of scheme theory as compared to tamer geometric structures (that subschemes are not determined by subsets, that functions are not determined by their values, etc) without adding the complications due to sheaves and gluing. I remember it took me a long time to realize this and when I did I lost some of my fear of schemes.

I fear that much of the apparent pathology in the theory of schemes comes from the presentation of schemes as locally ringed spaces rather than their presentations as sheaves of sets on the category of affine schemes (and therefore their presentations as the gros slice toposes they represent). For example, the reason the fibre product doesn't make any sense at all (even on the underlying set of the locally ringed space) is that Sch has a faithful (and full?) embedding into LRS. When we compute the fibre product of schemes as abstract sheaves, we compute it pointwise, which gives a right answer
–
Harry GindiJun 17 '10 at 14:59

27

Harry, if you're not sure whether the functor from schemes into locally ringed spaces is full, you should step back from the etale topos and related formalism and learn more basic things better (and be more humble for the present time about offering advice on how to think about or teach the subject).
–
BoyarskyJun 17 '10 at 15:56

21

Harry, you misunderstood my point: the fact that you needed to go back and check this basic fact from the beginning of the theory means you had not internalized it, and so is a reflection of a certain lack of experience on your part (it is one of the first things that one should learn in the theory of schemes, to connect it up with other geometric theories, etc.). You should consequently be more reserved in offering advice to others on how to teach or think about it. It is akin to a real analysis student who needs to go back and check whether or not the Intermediate Value Theorem is true.
–
BoyarskyJun 17 '10 at 16:44

13

Harry, it was not intended as a rebuke (which I interpret as a somewhat negative word). It was simply advice to be more reserved, in view of your somewhat limited experience in this area of mathematics. By all means discuss these ideas with your classmates, professors, etc. Just be less energetic about making suggestions on educational aspects until you have had more time to see where it all goes and how it is used and how more of the deep theorems are proved.
–
BoyarskyJun 17 '10 at 17:02

Victor Protsak suggests this, and I'll endorse it: the careful construction of the Grassmannian. This is a good example for 3 reasons. (1) It is extremely important. (2) It is a situation where it is both natural to work in local coordinates and with a global projective embedding, so students can practice transforming between the two perspectives. (3) It is small enough to do in full detail, but it usually isn't.

Ideally, this would include proving that the Grassmannian represents the functor of flat families of subspaces of a vector space. (Or quotient spaces, whichever you prefer.)

Sorry, but I do not understand how the Grassmannian motivates cohomology and base change?
–
Kevin H. LinJun 18 '10 at 15:15

1

Sorry Kevin, I should have been more explicit. First you understand the Grassmannian in terms of quotients of a free bundle, prettily by hand. But you might also want to think of it geometrically, as parametrizing $\mathbb{P}^k$s in $\mathbb{P}^n$, i.e. a special case of a Hilbert functor: a family is a closed subscheme of $\mathbb{P}^n$ over the base, flat, whose fibers are (linear) $\mathbb{P}^k$s. How might you turn it into the linear algebra problem? By pushing forward the restriction map for $\mathcal{O}(1)$. How do you know that the resulting things are locally free? Coho + base change!
–
Ravi VakilJun 18 '10 at 19:25

I haven't seen it mentioned yet, so let me suggest it (and I'll be curious to hear people's
responses): the theorem on formal functions.

In suggesting this, I am certainly taking full advantage of the fact that we are supposed to be discussing a year long course.

Let me now give justification (in case it is needed; I don't know how others will feel
about this suggestion).

First, my own philosophy is that an algebraic geometry course, even one focusing on the theory of schemes, should be about geometry. So I think that it is important to discuss some geometry, including the basic theory of curves (which is very pretty from the schemes point of view, since one gets the interaction between a more geometric picture, and the more valuation-theoretic function-field picture, by studying the interaction between the generic point and the closed points).

But the theory of curves is not enough concrete geometry for one year; I think some discussion of surfaces adds an enormous level of geometric understanding, just because the theory of surfaces is much closer to the theory of arbitrary dimensional varieties than the theory of curves is. At the same time, by doing some stuff with surfaces, one does a valuable service for many students in the class: pure geometers will certainly need to know this, but so will arithmetic geometers/number theorists, because a curve over a Dedekind domain behaves like a surface, and one studies bad reduction of curves using ideas from the theory of surfaces (blowing up, minimimal models, etc.). So even if one doesn't touch directly on the particularities of degenerations of curves (which, however, is also a topic of very general interest and importance!), by saying something about surfaces, one prepares the way.

Hartshorne Ch. V gives a really nice treatment of many of the basics of surface theory,
and the main tool he uses, beyond all the generalities of cohomology and sheaves, is the
theorem on formal functions: both in its application to Zariski's main theorem,
and to the proof of Castelnuovo's criterion. And these are both beautiful results, the kind
of results that would make a good capstone to a one year course.
(And they are also basic algebraic geometry knowledge --- the kind of things that you
would hope students know after taking a year of the stuff!)

Ding ding ding! Bells went off in my head when I read this, as this is a topic I very much wanted to hear people's opinions about. Matt, I agree with you that this is important, and in particular through ZMT and Castelnuovo's criterion. I also agree that this is fair game because we are talking about a year-long course, and this would be near the end. So I think it should be in if all possible. (Unlike, perhaps, formal schemes, which no one has brought up so far.) cont'd
–
Ravi VakilJun 20 '10 at 21:01

3

Ravi, proof of thm on formal fns in EGA is simpler than version in Hartshorne, and it works directly in the proper case (no mucking around with $\mathcal{O}(n)$'s, etc.). The argument is due to Serre, and it gives a stronger result than what is claimed in Hartshorne in the projective case. I wrote up a handout on it for the course I taught with Matt's help way back when. I can send you the .pdf for it if you don't have it buried in a filing cabinet somewhere. (As an aside, I think it would be a mistake to try to introduce formal schemes, though one could say what the point of it is.)
–
BCnrdJun 20 '10 at 23:05

1

@Ravi There's a resource you may or may not be aware of and I've posted it at Math Online:Last year,Micheal Artin taught a wonderful looking first course at MIT on algebraic geometry using his own notes and William Fulton's ALGEBRAIC CURVES.The only prerequisites were a year long algebra course based on the forthcoming second edition of his classic text.Not only are the notes themselves excellent as a model for what that preliminary "classical" course before yours should look like,he has some very insightful comments there on the teaching of AG. I think you'll find it quite useful.
–
The MathemagicianJul 16 '10 at 20:40

1

You do not <B>NEED</B> the theorem on formal functions to prove Castelnuovo's criterion. Smoothness of the target can be proved by considering the multiplicity of the image point. However, this approach to Castelnuovo's criterion is very close in spirit to Artin's "Algebraization of formal moduli II", cf. also Appendix B.3 of Hartshorne.
–
Jason StarrJul 28 '11 at 17:37

Why the Spec functor is a natural thing; this is not so clear (at least to me) from the definition in Hartshorne. Bas Edixhoven made me see the light by saying that Spec is adjoint to the global sections functor from locally ringed spaces to commutative rings: $\mathrm{Hom}_{\mathrm{Rings}}(A,\Gamma(X,{\cal O}_X))\cong\mathrm{Hom}_{\mathrm{LRS}}(X,\mathrm{Spec}(A))$. Exercise II.2.4 of Hartshorne asks you to prove this with locally ringed spaces replaced by schemes, but this is less clarifying.

Interesting follow-up: this question suggests that Spec is the right thing because "locally ringed spaces" are the right kind of geometric space. But people may not be initially convinced that locally ringed spaces are natural. Perhaps Spec could be defined first (hence your question still stands), and the locally ringed spaces next? It is a bit of a chicken-and-egg thing. (I'm undecided/agnostic about this.)
–
Ravi VakilJun 20 '10 at 20:57

I actually think that the Hilbert scheme should be mentioned (and, if possible, proved to exist and discussed) as early as possible. It serves as a good example of a moduli space, and it exists! Plus, the infinitesimal study of the Hilbert scheme allows some deformation theory to be discussed (at least, the deformations of projective schemes inside projective space) which also helps explain, algebro-geometrically, what the normal sheaf really controls. Add to this the fact that a lot of research relies on moduli spaces these days (In particular, I know that people care about Hilbert schemes of points, and, if some GIT for PGL can be covered, it'll let you actually construct $\mathcal{M}_g$, which finishes the classification of curves that's given in chapter 1 of Hartshorne, though this is a bit more.)

Because you'll be wanting things fundamentally scheme theoretic, the first part of Kollár's "Rational Curves on Algebraic Varieties" might be a good reference for this stuff.

Kevin, this opens up a can of worms, to nail down the compatibility of the coherent trace on ${\rm{H}}^n(\Omega^n)$ and the topological trace on ${\rm{H}}^{2n}(\mathbf{C})$ with respect to the "degeneration isomorphism" between them. I had a long conversation about it with Serre, and he was very disappointed with the literature. So when "mentioning" to someone that Serre duality is a refinement, one should also mention that there is real work needed to nail down the compatibilities. There's an old .pdf file about it in the "duality" part of my webpage; quite tricky to do rigorously.
–
BCnrdJun 18 '10 at 7:11

(Maybe this is a standard thing to do already but I think it's still worth mentioning:) A proof of Bezout's theorem via Hilbert polynomials of subschemes of $\mathbb P^N$.

Of course, this isn't fundamentally different than the proof in Hartshorne I.7, but in scheme language it is much much more natural, and might be the best motivation for allowing nilpotents in the structure sheaf.

This is extremely vague, but it's something I wish someone should have told me 5 years earlier: Since algebraic geometry is so rigid (few polynomials compared to many differentiable functions), we often have to deal with singularities. E.g. in many cases we can't make intersections transversal, or all interesting families (of certain types) have singular fibers. But since algebraic geometry is so rigid, we also have fairly good tools dealing with singularities, or with degenerate cases.

I think point 2 here is an extremely important one to make. One doesn't have to give a song and dance about it, but it should be clearly stated at the beginning of any algebraic geometry course. To students who don't yet have much experience with other kinds of geometry, it may not mean so much at the beginning, but as they mature, it will hopefully stay with them as a guide to the difference between algebraic geometry and other geometries. For those who are used to other geometries and want to learn algebraic geometry, I think it is one of the first things that should be pointed out.
–
EmertonJul 24 '10 at 0:27

1

It's also a nice heuristic to explain why methods like degenerating techniques, (virtual) localization, ... are so powerful (or why flatness is such an important notion).
–
Arend BayerJul 24 '10 at 2:05

Stalk-local detection of irreducibility on locally Noetherian schemes, which I prove directly here with no primary decomposition tricks. It helps with a lot of exercises, and intuition.

Sheafification of base-presheaves (presheaves defined only on a base of open sets). I see from your TOC that you cover the unique extension of base sheaves to sheaves as per Kevin Lin's answer (E-H's Proposition I-12).

When I took Arthur Ogus' algebraic geometry class, he was very insistent about teaching us this, and it really paid off for the remainder of the course, particularly in exercises. It categorically exclaims (pun intended) the credo always start with the affine opens, so one sees explicitly how special and critical they are to the theory.

The sheaf of meromorphic functions $\mathcal{K}_X$ on $X$ can be defined by sheafifying the naive base-presheaf $\mathcal{K'}(U)=Frac(\mathcal{O}(U))$ on the base of open affines. This formula doesn't define a base-sheaf on affines, and as Georges Elencwajg and BCnrd explain here, it doesn't even define a presheaf when applied to arbitrary opens. I suggest at least mentioning these three facts, to save people from re-wasting the time that I and many others have in wondering what the resulting sheaf looks like.

Locally representable means representable, i.e. if $F:Sch^{op}\to Set$ is a sheaf when restricted to a base of (Zariski) opens on every scheme, and $F$ has a covering by representable open subfunctors $F_i$, then $F$ is representable (very much along the lines of EGA 1 (1971), Chapter 0, Proposition 4.5.4). I advocate this because the work that goes into the proof is essentially the same work we inevitably do to prove fibered products of schemes exist, so it gives fibre products as a special case, but also offers up a rigorous-but-quick route to other constructions like global Spec and global Proj.

The general definition of quasicoherence and coherence for modules on local ringed spaces / non- locally Noetherian schemes... not as a gratuitous generality, but as a foreshadowing/reminder that presentations, not just surjections, are what make coherence work.

Basic Dedekind domain theory, along the lines of Lang's Algebraic Number Theory, chapter 1. I found curves and their divisors — even in characteristic 0 — impossible to understand until I read that.

Quasiseparatedness is something I'm glad to see you including, because using it explicitly is the key to a lot of proofs, so having it in mind as a word helps me remember how to do them.

Added by request: here's how I think about Serre's criterion. Call a rational function pretty good if it doesn't blow up in codim 1. Call it very good if it's actually well-defined in codim 1. Then a normal space is one for which pretty good rational functions are actually functions, whereas an S2 space only asks that very good rational functions are actually functions. To see the difference, look at x/(x+y) on {xy=0}, to see that the latter is not normal despite being S2. So how can normality fail -- how can f's value be ambiguous in codim 1? If there are 2 ways to approach some divisor -- non-R1ness.

Somewhat related to this: is the notion of Cohen-Macaulay something people really should see early on? Advantages: it doesn't take long. You get to see the Koszul complex. Then you get to see that the normal sheaf to a local complete intersection is locally free. There is a handy flatness theorem (very roughly, a map from CM to nonsingular is flat iff the fibers are equidimensional; and a flat map to a nonsingular has CM source). Then S2 could go here, perhaps later than it needs to be. Disadvantage: yet more definitions to clog up your brain.
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Ravi VakilJun 18 '10 at 0:05

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One of the things I regret is that none of my teachers ever taught me what "Cohen-Macaulay" means. Depending on what you do, it can show up pretty much first thing as you set out into the literature, with the assumption that you know it cold already.
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Charles SiegelJun 18 '10 at 4:16

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@Charles: wouldn't such an experience simply provide motivation to go back to the commutative algebra books or elsewhere to learn about it (if it wasn't learned when doing Serre duality)? I do agree that CM is a very good notion to see in a course, if time permits. But virtually everything I know about modern algebraic geometry I had to teach myself, often in the service of trying to understand other things which I cared about. But that's part of doing math: having to struggle with learning stuff on one's own, for which external motivation is always a good thing.
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BCnrdJun 18 '10 at 7:18

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BTW here's how I think about Serre's criterion. Call a rational function pretty good if it doesn't blow up in codim 1. Call it very good if it's actually well-defined in codim 1. Then a normal space is one for which pretty good rational functions are actually functions, whereas an S2 space only asks that very good rational functions are actually functions. To see the difference, look at x/(x+y) on {xy=0}, to see that the latter is not normal despite being S2. So how can normality fail -- how can f's value be ambiguous in codim 1? If there are 2 ways to approach some divisor -- non-R1ness.
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Allen KnutsonJun 20 '10 at 1:19

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@Emerton: normal => S2 => hyperplane sections are S1 => if each component has a reduced point, then the hyperplane section is reduced. Which I think is pretty cool. I used a similar implication in arxiv.org/abs/math/0306275 , namely that a generically reduced complete intersection is in fact reduced (really, that CM => S1).
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Allen KnutsonJun 21 '10 at 2:44

Oh my. Harry, I don't know what has led you to exert so much time on higher topos theory in lieu of getting more experience with schemes first (the motivating problems, the geometry, the insights from the etale topology, etc.), but you would benefit from seeking more guidance from experts at UM. The (very geometric!) inspiration for algebraic spaces comes from Artin approximation. Even Jacob Lurie learned the basic theory of schemes thoroughly (in a course run by me and Emerton, with no topoi but lots of balance of theory and examples) before going on to the etale site and stacks. Good luck.
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BCnrdJun 19 '10 at 23:52

If you decide to teach a more arithmetically flavoured algebraic geometry, students should be made aware that schemes over a ring $A$ are stranger than they might think.

For example $A$-rational points of $\mathbb P^n_A$ are far from being given by non-zero $(n+1)$-tuples of n elements of $A$ modulo tuples of invertible elements, but are described by rank-$n$ projective summands of $A^{n+1}$. More generally morphisms to projective space are described in terms of line bundles and their sections and might be seen as an interesting illustration of these concepts.

Incidentally, a sufficient reason for introducing a little arithmetic geometry is to have the pleasure of reproducing Mumford's incredibly enlightening drawing of the arithmetic surface $\mathbb A^1_{\mathbb Z}$ (in his Red Book), with its points having each a diferent personality and its curves. ( I concede that although Mumford's picture is beautiful, the artistic competition was not so great when he wrote his notes: the EGA's strongest point is not its illustrations...)

Georges: the common "surprise" about points of projective space or of affine space minus 0-section valued in a ring (or scheme) has always seemed best to explain by analogy with how the same issue comes up in differential geometry, or even alg. geom. using only varieties and not schemes. The meaning of a map from a manifold to real projective $n$-space works out exactly as with schemes, and likewise for affine space minus the 0-section, so it is good to stress to students that none of this is peculiar to working with schemes or is a phenomenon special to the "arithmetic" case.
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BCnrdJun 18 '10 at 13:21

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I strongly disagree with you about EGA. Every single picture in EGA is incredibly enlightening, and beautifully rendered. :-) [Anyone who has looked at EGA will realize I'm actually agreeing with Georges, but that my "Every single picture" comment is also true...]
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Ravi VakilJun 18 '10 at 13:53

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I am very proud of your endorsement, Ravi: thank you. Contrariwise, my heart missed a beat when I read your sentence "I strongly disagree with you about EGA" but fortunately, reading on, I realized that the rich resources of the empty set were coming to my rescue :-)
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Georges ElencwajgJun 18 '10 at 14:15

Perhaps this should be attached to Charles Siegel's answer about the Hilbert scheme, but some concrete examples of degenerating flat families could be helpful. Some easy examples include conics turning into a fat line, skew lines colliding to produce an embedded point, and pairs of points on a line colliding to become fat. There are some nice relationships between these objects and families of constant coefficient linear differential equations via spectral schemes, e.g., the colliding points example says something about the behavior of solutions to $(\frac{d}{dz} - a)^2 - \lambda^2 = 0$ as $\lambda$ hits zero.

The point is that many definitions in algebraic geometry are basically obtained by taking definitions from topology or algebra, translating them into "purely category theoretic language" and then using that definition as a substitute in the category of schemes.

In particular I unravel the definition of a separated morphism:

"A seperated morphism of schemes is one where the image of the diagonal is closed."

If we just replace "schemes" with "topological spaces", then this property for spaces says (after a little definition chasing)

"Any two distinct points which are identified by the morphism can be separated by disjoint open sets in the domain"

Thus a space is Hausdorff as a topological space iff the unique map to the one point space is separated. Before I worked through this I had no real reason to believe that separated morphisms were a natural concept. Why don't people ever talk about the topological analogue?

Another point of much confusion for me was the definition of derived functor cohomology. Why should we care about injective resolutions? Anton gives a great answer here:

Anton's line of thought is also beautifully developed in Gunter Harder's book "Lectures on Algebraic Geometry 1". The quick and dirty version is that cohomology should have nice properties (ses gives rise to les, etc) and acyclic resolutions compute cohomology. Hey! Injective objects are always acyclic (this is reasonable because they make ses's split). Thus injective resolutions are a nice generic thing to use.

I agree with your frustration; certainly students tell younger students this fact. (And I also included it as an exercise.) I don't think Hartshorne's discussion of separatedness is representative; the description in EGA is clearer (although of course sans motivation). I haven't checked, but perhaps only Hartshorne (among the major references) uses the valuative criterion to prove things that can be more easily proved by hand.
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Ravi VakilJun 18 '10 at 18:48

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On a related note, I conjecture that the reason that recent generations learn the valuative criterion so early is that Hartshorne does this. Inflammatory comment: It is not clear to me that anything one might reasonably see in a first course can be more easily proved with the valuative criterion than directly, taking into account the cost of proving the valuative criterion. Any counterexamples? (Although certainly the statement is worth seeing early. But the proof can go, to make room for other things.)
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Ravi VakilJun 18 '10 at 18:51

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OK Ravi, I'll bite: doesn't the proof of universal closedness of projective space (over Z) via val. crit. demonstrate the elegance of functorial criteria (in comparison with elimination theory)? There's also something likewise cool about using it to prove that the map from Grassmannian to projective space is closed immersion (esp. comparing it against the more explicit traditional proof, which is also concrete and worth seeing). Basically, it provides simple examples of the power of functorial criteria (if not overdone!). That is surely something to be appreciated in a first course.
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BCnrdJun 18 '10 at 19:17

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@Ravi For the record,from most of the AG graduate students who have both been in your class and are using the posted notes online that I've spoken to,your notes are proving an invaluable resource for them in the learning of schemes and modern AG. Many of them in fact have dumped Hartshorne in favor of your notes.Thier continued evolution and availabilty on the web has gained a growing grass-roots support.I think I can speak for all of them when I say this ongoing project is a noble undertaking and it's continuation is fervently hoped for.
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The MathemagicianJul 16 '10 at 20:46

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On the derived-functors versus Cech-cohomology-only question: I think an interesting middle ground is to do cohomology only for quasi-coherent sheaves on separated schemes. One can construct injective resolutions via an affine covering and I-twiddles of injective modules I for each open set. Then one can prove e.g. the agreement with Cech cohomology without ever using words like "flasque" or "$O_X$-modules". (My feeling is that at least some of the obfuscation comes from the fact that Hartshorne's approach leaves the category of quasi-coherent sheaves before taking injective resolutions.)
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Arend BayerJul 24 '10 at 2:33

I am surprised that no one mentioned this so far; I am only imagining that everyone thought it so natural that it escaped their mind.

Most "standard courses" would be following Hartshorne's book, I assume. It is a great loss that this book does not mention the "functor of points" view at all. It would maybe take 10 or 15 minutes to state and prove the Yoneda's lemma, and a little more time to mention the functor of points and the advantage of this point of view for applications to arithmetic geometry(points with values in a certain ring, base change, etc.), and more importantly for moduli problems. One could also give a definition of a fine moduli space and coarse moduli space, and as examples just mention the the moduli space of curves with marked points(but without proofs, of course).

A small suggestion : the deformation to the normal cone is a nice construction that I would have liked to see in a first course. It illustrate the use of blow-ups, the degeneration of a family with constant fibers (an highly non-obvious concept the first times you see it) and how important intuitions from differential geometry - tubular neighbourhoods - have a non-trivial translation to algebraic geometry.

In reference to why the spec functor is a natural thing, (low tech answer): isn't this essentially what the nullstellensatz says? Or rather it generalizes the nullstellensatz. I.e. spec is a good thing because it lets you make a construction that gives you some "geometry" associated to a given ring.

Perhaps the main thing beginners should learn about schemes is that they are needed. I.e. schemes should be motivated. In books which try to restrict to varieties such as Shafarevich's BAG, schemes still raise their heads sometimes unnoticed. E.g. Shafarevich states in chapter I sections 4.4 and 6.4 that the set of hypersurfaces of given degree in a given projective space are parametrized by a projective space, which is not true unless one considers more than the variety defined by a polynomial.

If one is guided on what to include by the section headings of chapter 2 of Mumford's red book, in addition to fields of definition and the functor of points, one finds there a section called specializations, which also contains one of his exotic illustrations.

Even in a classical book like Walker's algebraic curves, schemes arise when studying singularities. The tangent cone to a cuspidal plane curve requires more structure than a variety. Even the fundamental theorem of algebra does not count the roots of a polynomial correctly unless multiplicities are considered.

Some of these examples require only cycles or divisors rather than schemes, but more general tangent cones should provide more general schemes. One can also consider the problem of varieties varying in families and try to fill in something over the limit point of the parameter space. Sometimes non reduced objects will force themselves on us.

The best motivation for differentials may be learning the classical Riemann Roch theorem for curves.

Of course this is probably obvious and taken for granted by most people, but it seemed worth mentioning as a guide to choosing first examples of schemes. I.e. we should not take schemes for granted and choose what to teach based solely on the needs of experts, but we should assume that schemes may be quite strange to beginners and spend some effort showing that they are natural.

I think that base change is a very important and subtle idea which should certainly be included in a first course. In particular, one should discuss properties that are stable under base change and those that are not.

In a similar vein, in discussing cohomology, the difference between the coefficients of the motive and the base should be emphasized. This was confusing to me as I learned the subject.

Well, I think that I'll elaborate with an example. If $X=$Spec$F$ is a variety over $\mathbb{Q}$ and $F$ is a number field then $H^0_B(X(\mathbb{C}),\mathbb{Q})$ is isomorphic to the group ring $\mathbb Q[G]$ where $G$ is the Galois group of $F$ over $\mathbb{Q|$. If I would like to decompose this into the irreducible representations of $\mathbb{Q}$, then I need to extend \em{coefficients} to a field $E$ over which the idempotents are defined. So I would be looking at $H^0_B(X(\mathbb{C}),E)$. If I wanted to look at a subgroup of $G$, I would need to change the base.
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Johnson-LeungJun 18 '10 at 12:25

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Another example: an $\ell$-adic sheaf on $X$ is a motive over $X$ with coefficients in $\mathbf{Q}_{\ell}$ (or rather a realization of one). The most interesting cases are when there's a relationship between the coefficients and the base, e.g. Hodge theory and $p$-adic Hodge theory.
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JBorgerJun 18 '10 at 13:50

That isn't really an answer to the question - I don't think it's necessary in a first course, but I do think resolution should be rotated in on a regular basis, which requires the annual core to be small enough to make room for it. I think it's valuable not just to teach the material somewhere in the curriculum, but to put it in an introductory course, to emphasize that it is elementary and not impossibly difficult. Also, to contrast with the Grothendieck-flavored majority.

Do you mean a proof of resolution of singularities? (And I presume you mean in characteristic $0$.) At Brown, there was a topics course on this, using Koll\'ar's explanation; but it took a semester. Somehow I find resolution of singularities harder than most of Grothendieck's foundations.
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Ravi VakilJun 18 '10 at 0:11

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As far as learning blowups goes, a good (albeit tedious!) exercise that really got me comfortable with them was resolving the ADE surface singularities.
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Charles SiegelJun 18 '10 at 15:37

1.$ $ This is really about commutative algebra more than algebraic geometry as such, but something I found incredibly frustrating for a while was what to do when I need to compare $M \otimes_A N$ with $M \otimes_B N$. I finally discovered the following illuminating lemma:

If $M$ and $N$ are $B$ modules, then for every ring homomorphism $A \to B$, there is a natural map $M \otimes_A N \to M \otimes_B N$. Moreover, this map is an isomorphism for all $M, N$ iff it is an isomorphism for $M = N = B$ iff $A \to B$ is an epimorphism of rings.

In particular, the last condition holds if $B$ is obtained from $A$ by some combination of localization and taking a quotient ring, or if $\operatorname{Spec} B \to \operatorname{Spec} A$ is any kind of immersion. The same "abstract nonsense" shows that if $Z \to Z'$ is a monomorphism of schemes (in particular, any kind of immersion), then the product of two $Z$-schemes over $Z$ is naturally isomorphic to their product over $Z'$.

2.$ $ I found the usual description of gluing schemes and morphisms (i.e., requiring things to agree on $U_i \cap U_j$) frustrating to use sometimes, because in general, $U_i \cap U_j$ might not be affine even if $U_i$ and $U_j$ both are. To glue morphisms only, one can require that the morphism be defined on every set of an open cover, such that whenever $x \in U \cap V$, then $x$ has a neigborhood $W \subset U \cap V$ such that $f_V|W = f_U|W$.

For gluing schemes, one can use a commuting poset of open immersions. Given such a diagram, with objects $\{U_i\}$, there exists a scheme $W$, together with open immersions $U_i \to W$ commuting with the diagram, such that the $U_i$ cover $W$, and $x \in U_i$, $y \in U_j$ map to the same point in $W$ iff they may to the same point in some $U_k$. When this is combined with the statement on gluing morphisms, one sees that $W$ is actually the colimit of the diagram; and, in fact, the statement that "any such diagram has a colimit" more or less encapsulates both glueing schemes and glueing morphisms. Realizing this was also the first time I felt like I understood colimits.

For a more streamlined, if less general, version of the above, one can use a version of the cocycle condition with $U_i \cap U_j$ replaced by a cover of $U_i \cap U_j$ by simultaneously distinguished affines, assuming the cover $\{U_i\}$ is by open affines.

In either formulation, this combines with the previous point to give a very quick construction of the fibre product: simply take the colimit of the diagram consisting of maps $$\operatorname{Spec} (A \otimes_C B)_{f \otimes g} \to \operatorname{Spec} A \otimes_C B$$
such that the images of $\operatorname{Spec} A$ and $\operatorname{Spec} B$ lie in $\operatorname{Spec} C$. (If these images in fact lie in a distinguished open subset $C_h$, we get the tensor product over that for free by point 1.) Of course, one still has to verify that this colimit behaves as desired; but this is not hard using the more general "gluing morphims" to show existence and uniqueness.

Note: if it's not clear already, my perspective is that of a student rather than an expert.

First off, I wanted to commend you on this whole project, Ravi. Algebraic geometry and the theory of schemes is a notoriously difficult subject to internalize for any advanced student and it's clear you've given a lot of serious thought on how to make it more digestible. I've browsed the old version of the notes and found them very readable and highly thought out. I firmly support this project and hope it goes through many revisions and drafts, evolving into a future classic. Algebraic geometry is a subject I haven't seriously begun broaching yet and I hope to use one of the newer versions when ready,

Secondly-I sympathize with your hesitancy to convert them into a book. What you might consider is creating an online text that will constantly be revised and will never be in "final" form. My old biochemistry professor Burton Tropp did this for many years and it worked out for him very well: The first edition WAS published, but all subsequent editions (and there was nearly a dozen before he retired last year) were online and subject to constant revision and improvement. I think this kind of format will work very well for you.

Thirdly -- history is so important in learning a new,conceptually difficult field. Some good historical notes would make the notes a lot more interesting to read no matter how good the exposition is. Students want to know how they came up with this crazy stuff -- if you know how the original source authors came up with these concepts and why, it'll make it a lot easier to not only internalize, but also to form thier own opinions on the subject.

Fourthly -- I think inserting references and research assignments relying on significant papers, such as Grothendiek's original schemes paper -- will give your students some much needed research experience in a very active field. These are advanced students and the more such experience they get,the better off they'll be.

Lastly -- I wanted to commend your humility and determination in asking other mathematicians and students for opinions and input on this project. It shows how committed you are to this project and experts should be chomping at the bit to give you thier feedback and opinions. I would, but my lack of expertise precludes that. Hopefully those with much more knowledge then I will jump at the chance to assist you with this wonderful project.

Good luck with this exciting project and looking forward to future versions!!!

For your third point Dieudonne has written "History of Algebraic Geometry", which starts at the very beginning (having separated the development of the subject into epochs), is a pleasure to read, and leads the reader to the modern problems. Unfortunately it appears to be rare, and I'm not sure if one would be able to distribute scanned pages among students.
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pmoduliJun 17 '10 at 18:52

At a relatively late point in the course, I believe that the idea of descent should be explained, with two examples: Zariski-descent, or gluing, and faithfully flat descent.
The latter should then be applied, for example to prove that some examples of functor of points are representable.

As David and Anweshi told before, think it could be very interesting to deal with functor of points, with main example being subfunctors of Grassmannians. I would make some general statements on functor of points (Yoneda lemma, definition of functor of points, vector bundles) and then begin to study as soon as possible classical examples, such as Grassmanians, Severi-Brauer varieties and their tautological vector bundle, varieties of flag of subspaces...

Finally it would lead to a glimpse on group schemes and algebraic groups.